Integral Equations and Operator Theory

, Volume 12, Issue 2, pp 163–210 | Cite as

The Feynman-Kac formula with a Lebesgue-Stieltjes measure: An integral equation in the general case

  • Michel L. Lapidus
Article
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Abstract

Let u(t) be the operator associated by path integration with the Feynman-Kac functional in which the time integration is performed with respect to an arbitrary Borel measure η instead of ordinary Lebesgue measurel. We show that u(t), considered as a function of time t, satisfies a Volterra-Stieltjes integral equation, denoted by (*). We refer to this result as the “Feynman-Kac formula with a Lebesgue-Stieltjes measure”. Indeed, when n=l, we recover the classical Feynman-Kac formula since (*) then yields the heat (resp., Schrödinger) equation in the diffusion (resp., quantum mechanical) case. We stress that the measure η is in general the sum of an absolutely continuous, a singular continuous and a (countably supported) discrete part. We also study various properties of (*) and of its solution. These results extend and use previous work of the author dealing with measures having finitely supported discrete part (Stud. Appl. Math.76 (1987), 93–132); they seem to be new in the diffusion (or “imaginary time”) as well as in the quantum mechanical (or “real time”) case.

Keywords

Integral Equation Time Integration Path Integration Borel Measure Imaginary Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Birkhäuser Verlag 1989

Authors and Affiliations

  • Michel L. Lapidus
    • 1
  1. 1.Department of MathematicsThe University of Georgia Boyd Graduate Studies Research CenterAthensUSA

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